The Evolution of Dynamics: Vibration Theory from 1687 to 1742

192 Pages·1981·8.836 MB·English

by  John T. Cannon, Sigalia Dostrovsky (auth.)

Studies
in the History of Mathematics and
Physical Sciences
6
Editor
G. J. Toomer
Advisory Board
R. Boas P. Davis T. Hawkins
M. J. Klein A. E. Shapiro D. Whiteside
John T. Cannon
Sigalia Dostrovsky
The Evolution of Dynamics:
Vibration Theory from
1687 to 1742
With 10 Illustrations
Springer-Verlag
New York Heidelberg Berlin
JOHN T. CANNON
and
SIGALIA DOSTROVSKY
155 Fairfield Pike
Yellow Springs, Ohio 45387 jUSA
AMS Subject Classifications: 01A45, 01A50, 73-03, 73 D30
Library of Congress Cataloging in Publication Data
Cannon, John T.
The evolution of dynamics: vibration theory from 1687 to 1742.
(Studies in the history of mathematics and
physical sciences; 6)
Bibliography: p.
Includes index.
1. Vibration-History-17th century.
2. Vibration-History-18th century.
I. Dostrovsky, Sigalia. II. Title.
III. Series.
QA865.C36 531'.32 81-14353
ISBN-13: 978-1-4613-9463-1 AACR2
© 1981 by Springer-Verlag New York Inc.
Softcover reprint ofthe hardcover 1st edition 1981
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag, 175 Fifth Avenue, New
York, New York 10010, U.S.A.
9 8 7 6 5 432 1
ISBN-13: 978-1-4613-9463-1 e-ISBN-13: 978-1-4613-9461-7
DOl: 10.1007/978-1-4613-9461-7
Preface
In this study we are concerned with Vibration Theory and the Problem
of Dynamics during the half century that followed the publication of
Newton's Principia. The relationship that existed between these subject!!
is obscured in retrospection for it is now almost impossible not to view
(linear) Vibration Theory as linearized Dynamics. But during the half
century in question a theory of Dynamics did not exist; while Vibration
Theory comprised a good deal of acoustical information, posed definite
problems and obtained specific results. In fact, it was through problems
posed by Vibration Theory that a general theory of Dynamics was motivated
and discovered.
Believing that the emergence of Dynamics is a critically important link
in the history of mathematical science, we present this study with the
primary goal of providing a guide to the relevant works in the aforemen
tioned period. We try above all to make the contents of the works readily
accessible and we try to make clear the historical connections among many
of the pertinent ideas, especially those pertaining to Dynamics in many
degrees of freedom. But along the way we discuss other ideas on emerging
subjects such as Calculus, Linear Analysis, Differential Equations, Special
Functions, and Elasticity Theory, with which Vibration Theory is deeply
interwound. Many of these ideas are elementary but they appear in a
surprising context: For example the eigenvalue problem does not arise in
the context of special solutions to linear problems-it appears as a condition
for isochronous vibrations.
Although mathematical thought differs in different ages, mathematics
itself has a coherence that transcends time .. Thus it provides a powerful
tool with which to grasp modes of thought from former times. From an
immersion in the details of mathematical arguments, one can gather enough
precise understanding to be able to enter into the domain of the intuitive.
Therefore we befieve that our study not only describes a link in the evolution
of a specific subject but also that it assists in the attainment of a feel for
physics in the age of Newton and the Bernoullis.
In spite of its evident importance, dynamics in the first half of the
eighteenth century has been largely neglected. This is the period of late
Newton and early Euler; thus it lies in the shadow of great brilliance coming
from both before and after. For example, Euler was a central figure; but
vi Preface
his works from the period go with little notice because he later reworked
everything in a form and from a point of view that have become generally
familiar. Thus Truesdell's notes on Euler were pioneering works. 1 Truesdell
emphasized the fact that the idea of dynamical equations was slow to
emerge; furthermore, he provided a basic indication of the contents of a
vast number of papers, induding most of the papers considered in the
present study. We gratefully acknowledge our indebtedness to his notes.
Yellow Springs J.e. and S.D.
April 1981
1 Truesdell [1, 2].
Table of Contents
1. Introduction . . 1
2. Newton (1687)
9
2.1. Pressure Wave 9
2.2. Remarks . . 13
3. Taylor (1713) 15
3.1. Vibrating String 15
3.2. Absolute Frequency 19
3.3. Remarks . . . 20
4. Sauveur(1713) 23
4.1. Vibrating String 23
4.2. Remarks . . . 26
5. Hermann (1716)
28
5.1. Pressure Wave 28
5.2. Vibrating String 30
5.3. Remarks . . 31
6. Cramer (1722) 33
6.1. Sound 33
6.2. Remarks . 35
7. Euler (1727) 37
7.1. Vibrating Ring 37
7.2. Sound 43
8. Johann Bernoulli (1728) 47
8.1. Vibrating String (Continuous and Discrete) 47
8.2. Remark on the Energy Method ..... 52
viii Contents
9. Daniel Bernoulli (1733; 1734); Euler (1736) 53
9.1. Linked Pendulum and Hanging Chain 53
9.2. Laguerre Polynomials and 10 58
9.3. Double and Triple Pendula 60
9.4. Roots of Polynomials 61
9.5. Zeros of 10 ...... . 63
9.6. Other Boundary Conditions 64
9.7. The Bessel Functions Iv 66
10. Euler (1735) ..... 70
10.1. Pendulum Condition 70
10.2. Vibrating Rod 73
10.3. Remarks ..... 75
11. Johann II Bernoulli (1736) 77
11.1. Pressure Wave 77
11.2. Remarks 80
12. Daniel Bernoulli (1739; 1740) 83
12.1. Floating Body 83
12.2. Remarks 88
12.3. Dangling Rod 89
12.4. Remarks on Superposition 91
13. Daniel Bernoulli (1742) 93
13.1. Vibrating Rod ...... . . . . 94
13.2. Absolute Frequency and Experiments 99
13.3. Superposition ......... . 102
14. Euler (1742)
104
14.1. Linked Compound Pendulum 104
14.2. Dangling Rod and Weighted Chain 107
15. Johann Bernoulli (1742) 110
15.1. One Degree of Freedom 110
15.2. Dangling Rod 111
15.3. Linked Pendulum I 114
15.4. Linked Pendulum II 121
Contents ix
Appendix: Daniel Bernoulli's Papers on the Hanging
Chain and the Linked Pendulum 123
Theoremata de Oscillationibus Corporum . 125
De Oscillationibus Filo Flexili Connexorum 142
Theorems on the Oscillations of Bodies . . 156
On the Oscillations of Bodies Connected by a Flexible Thread 168
Bibliography 177
Index 182
1. Introduction
Before the middle of the eighteenth century, no-one had any notion
that "Newton's Second Law" could be used as the basis for a dynamical
description of a mechanical system that has several degrees of freedom.
Yet by name and by tradition this notion is associated with the Principia
of 1687 and it is commonly supposed that Newton, at least, knew that his
"Second Law" described in principle the motions of a mechanical system.
What is true, is that a lot was understood before the middle of the eighteenth
century about the dynamics of systems in one degree of freedom (as well
as about a few systems having special symmetries in a few degrees of
freedom) and Newton was indeed the greatest master of the period. Thus
the common supposition about Newton's understanding of dynamics is
created by ignoring the enormous distinction that separates mechanical
systems in one degree of freedom from those in more than one or, as we
will say, many degrees of freedom. On a popular level, this distinction has
the sound of a mere technicality, concerned only with the details of com
plexity, and it may be for this reason that it has been so commonly ignored.
One might wish for an adjective of moment to describe the difference
encountered when there are many degrees of freedom. (One can talk about
the dimension of the manifold of configurations; but in the present work,
dimension will refer only to physical space.) The purpose of the present
study is to follow in detail those works that dealt with the dynamics of
systems in many degrees of freedom up to the year 1742 (apart from the
cases, like the central force problem, that were handled because of their
special symmetries).
Thus, to grasp how it was that "Newton's Second Law" had a different
appearance before the middle of the eighteenth century, one should distin
guish between systems in one and in many degrees of freedom. We will
incorporate in our terminology a second distinction of a different type
which refers to' a conceptual limitation. "Newton's Second Law" in its
modern sense will be referred to as the momentum principle. Thus the
momentum principle (together with the corresponding principle for angular
momentum) entails the idea that a system of equations determines a
system's motion for given initial conditions. The equations require that one
specify forces even for states that are never reaijied in particular motions of
interest. But "Newton's Second Law," as it was understood before the
2 1. Introduction
middle of the eighteenth century, will be referred to as the momentum law.
The momentum law, then, is to be understood as a condition on a particular
motion of a mechanical system: the mass of an element of the system
multiplied by its acceleration must be equal to the force which that element
experiences during a given motion of the entire system. This is a condition on
a given motion and it is not a system of dynamical equations. In practice, the
momentum law is a consistency condition that typically leads to a constant
of a particular motion such as the period of an oscillation. We will see this
in examples, beginning with Newton's own work which we will discuss in
Chapter 2.
In the early eighteenth century, functional notation was used only in
very limited circumstances. In recognizing that certain concepts have a
vastly different appearance when a formulation from the point of view of
function theory is not available, one obtains a third distinction that helps
to clarify the early eighteenth century perspective on dynamics. For
example, if the momentum law had been formulated functionally, the force
on each element would have been given as a function of the state of the
system, and the momentum principle would have been quickly noticed.
That is, the difference between the momentum law and the momentum
principle can be maintained only as long as there is no general freedom
or incentive to make functional formulations.
In the case of a single degree of freedom, there was not a significant
limitation to giving a functional formulation of the momentum law and, in
fact, there was, in this case, occasional use of the momentum principle in
the early eighteenth century. But this use was not sufficient to suggest
seriously that a momentum principle could be used as a basis for dynamics.
Actually, the momentum principle played only an auxiliary role even for
the cases of one degree of freedom since, for a conservative system, it was
written in the form mv dv = F dy and it was immediately integrated to
obtain what we would call conservation of energy. Since the most typical
problem of this type concerns a particle constrained to a curve in the
gravitational field, conservation of energy itself was the most natural starting
point and,it sufficed for dynamics.
The question of functional formulations arose also in connection with
techniques of calculus. In fact, .during the eighteenth century, geometrical
calculus gradually gave way to functional calculus. The distinction between
the two calculi is clear in the following example: the equation dy =
{3.J (c2 - y 2) dt can be understood geometrically through the construction in
Figure 1.1'with its implied limiting situation. It is understood functionally if
one sets y = c sin ({3t + 8) with the sine understood as a function. Taylor
(Chapter 3) and Johann Bernoulli (Chapter 8) discovered that in its funda
mental mode, a vibrating string has the shape of a sine function-though
they did not express it this way. They expressed it geometrically and they
did not discover that also'~ the higher modes the string has the shape of
a sine function. That is, from the functional point of view, one naturally